Either disappointing a part of the party that has proved only too willing to rebel, or implementing a policy that risks real damage to their reputation for economic competence.
May, 61, is surrounded by people who dont support her but dont want her to quit. With Brexit, shes implementing a policy she opposed but will be her legacy.
And if her position seems both unsustainable and inescapable, its the same for the party she took over after last years in-or-out EU referendum cost her predecessor his job.
The Conservatives are openly discussing throwing May overboard. Yet their problems run deeper than one person. The party is split not just on the answer to the Brexit question, but on what the question actually is.
On one side there are those like Chancellor of the Exchequer Philip Hammond, for whom Brexit is a trade-off between separation from the EU and economic damage.
Then there are those such as Foreign Secretary Boris Johnson who argue that the economic advantages are greater the further and faster Britain is removed from the bloc.
The Conservatives consider themselves the party of government in Britain, safely managing the finances rather than getting caught up in too much political ideology.
The danger for the Conservative Party is that while it settles its individual battles in public, people start questioning if theyre fit for power, said Craig Oliver, a former director of communications for Mays predecessor, David Cameron.
If the civil war wasnt bad enough, Labour under Jeremy Corbyn, 68, has turned into a party sniffing at power again.
May is kept in place by a combination of factors: The lack of a replacement who is acceptable both to the pro-EU and anti-EU factions of her party; the fear that her fall would precipitate another election; and the hope that before shes thrown overboard, May can absorb some of the political damage likely to come when Britain leaves the EU.
Even should she find a Brexit position her Cabinet could agree on, she then has to get the members of the EU to agree to it. If they dont, she has to sell whatever they do agree to her party, and deal with the wrath of voters if the result is economic damage.
This is the sharpest version of a 200-year-old division in the Conservative Party, between nationalism and economic liberalism, said Stewart Wood, a Labour member of the House of Lords.
Any time May puts any flesh on the bones of a strategy, she loses support from one side or the other. Theres no solution.